1 00:00:00,005 --> 00:00:01,007 - [Instructor] In this chapter, 2 00:00:01,007 --> 00:00:04,001 you will explore numerical methods. 3 00:00:04,001 --> 00:00:05,007 These methods are great to use 4 00:00:05,007 --> 00:00:08,005 when traditional techniques to solve differential equations 5 00:00:08,005 --> 00:00:10,005 are inadequate. 6 00:00:10,005 --> 00:00:14,009 Let's begin by understanding what numerical methods are. 7 00:00:14,009 --> 00:00:17,002 So far, you have explored various techniques 8 00:00:17,002 --> 00:00:19,000 to solve differential equations, 9 00:00:19,000 --> 00:00:24,000 and these techniques focus on finding an exact solution. 10 00:00:24,000 --> 00:00:25,009 But what do you do if these techniques 11 00:00:25,009 --> 00:00:30,003 are extremely difficult or impossible to use? 12 00:00:30,003 --> 00:00:32,007 Instead of finding an exact solution, 13 00:00:32,007 --> 00:00:35,001 you can simply find an approximation 14 00:00:35,001 --> 00:00:38,003 utilizing numerical methods. 15 00:00:38,003 --> 00:00:41,001 Numerical methods are computational techniques 16 00:00:41,001 --> 00:00:45,002 utilized in mathematics to find approximate solutions. 17 00:00:45,002 --> 00:00:46,009 These methods are extremely helpful 18 00:00:46,009 --> 00:00:49,004 when you are unable to find a solution 19 00:00:49,004 --> 00:00:51,008 through traditional analytical techniques 20 00:00:51,008 --> 00:00:55,009 like the ones explored prior in this course. 21 00:00:55,009 --> 00:00:58,004 Numerical methods often heavily rely 22 00:00:58,004 --> 00:01:03,003 on calculators and computers to calculate their results. 23 00:01:03,003 --> 00:01:06,006 It is common to have software packages built 24 00:01:06,006 --> 00:01:08,006 for numerical methods 25 00:01:08,006 --> 00:01:11,006 in fields such as physics and engineering, 26 00:01:11,006 --> 00:01:14,000 where many key problems need to be solved 27 00:01:14,000 --> 00:01:16,005 with these approximations. 28 00:01:16,005 --> 00:01:19,004 I will not show you how to code with these techniques 29 00:01:19,004 --> 00:01:20,005 in this course, 30 00:01:20,005 --> 00:01:24,008 but I recommend exploring it more on your own. 31 00:01:24,008 --> 00:01:26,009 Numerical methods are great to use 32 00:01:26,009 --> 00:01:29,001 with more complex differential equations, 33 00:01:29,001 --> 00:01:33,008 especially when they are non-linear or have a high order. 34 00:01:33,008 --> 00:01:36,003 They can help solve both initial value problems 35 00:01:36,003 --> 00:01:39,006 and boundary value problems. 36 00:01:39,006 --> 00:01:43,002 They are also used in various practical applications 37 00:01:43,002 --> 00:01:44,006 like the ones mentioned earlier, 38 00:01:44,006 --> 00:01:49,002 such as physics, engineering, and biology. 39 00:01:49,002 --> 00:01:50,007 The key thing to remember 40 00:01:50,007 --> 00:01:54,003 is that numerical methods are an approximation. 41 00:01:54,003 --> 00:02:00,002 This means the result you'll get will not be 100% accurate. 42 00:02:00,002 --> 00:02:03,003 There will always be some certain degree of error, 43 00:02:03,003 --> 00:02:05,005 including when solving the same problem 44 00:02:05,005 --> 00:02:07,000 using different devices 45 00:02:07,000 --> 00:02:10,002 such as a computer versus a calculator 46 00:02:10,002 --> 00:02:13,001 or two different computers. 47 00:02:13,001 --> 00:02:15,005 Much of the time these errors are small, 48 00:02:15,005 --> 00:02:18,002 but be careful since they can quickly increase 49 00:02:18,002 --> 00:02:21,008 depending on the complexity of the equation being solved 50 00:02:21,008 --> 00:02:25,005 and the technique being used. 51 00:02:25,005 --> 00:02:26,009 You'll also want to make sure 52 00:02:26,009 --> 00:02:29,007 you account for round off and truncation errors 53 00:02:29,007 --> 00:02:34,008 when calculating with these numerical methods. 54 00:02:34,008 --> 00:02:37,006 There are a few other considerations to keep in mind 55 00:02:37,006 --> 00:02:41,005 when working with numerical methods. 56 00:02:41,005 --> 00:02:43,005 First, note that these techniques 57 00:02:43,005 --> 00:02:46,000 tend to be computationally heavy, 58 00:02:46,000 --> 00:02:48,000 meaning that there's sometimes a trade-off 59 00:02:48,000 --> 00:02:53,008 between accuracy, step size, and computational cost. 60 00:02:53,008 --> 00:02:56,001 Some of these methods like Euler's method 61 00:02:56,001 --> 00:02:58,001 can become unstable, 62 00:02:58,001 --> 00:03:00,007 so you may need to use very small step sizes 63 00:03:00,007 --> 00:03:04,003 in order to manage the stability. 64 00:03:04,003 --> 00:03:07,002 There are many numerical approximation methods 65 00:03:07,002 --> 00:03:08,009 out there to explore. 66 00:03:08,009 --> 00:03:12,006 In this course, you'll focus on learning Euler's method, 67 00:03:12,006 --> 00:03:16,003 Improved Euler's method, Runge-Kutta method, 68 00:03:16,003 --> 00:03:18,006 stability and convergence, 69 00:03:18,006 --> 00:03:20,007 and various practical applications 70 00:03:20,007 --> 00:03:25,002 that will be explored in the last video of this chapter. 71 00:03:25,002 --> 00:03:27,005 Numerical methods can be extremely useful 72 00:03:27,005 --> 00:03:29,003 for finding approximate solutions 73 00:03:29,003 --> 00:03:31,009 for complex differential equations. 74 00:03:31,009 --> 00:03:33,000 The goal of this chapter 75 00:03:33,000 --> 00:03:35,000 is to help you understand these methods 76 00:03:35,000 --> 00:03:36,004 and how they operate, 77 00:03:36,004 --> 00:03:39,003 but note that you can expand upon using these methods 78 00:03:39,003 --> 00:03:42,005 with coding and various software packages. 79 00:03:42,005 --> 00:03:46,000 Let's begin by exploring Euler's Method.